# Signatures

Nocturne uses a variant of Schnorr signatures over Baby Jubjub. The secret key is derived from the spending key $$sk$$, and the public key is the spending public key $$\text{PK}$$. 

Spelled out, signing takes as input a message hash $$m$$ and proceeds as follows:

1. $$h \leftarrow \text{SHA512}(sk)$$
2. $$s \leftarrow h\[0:32]$$(derive the signing secret key from the spending key)
3. $$x \leftarrow h\[32:64]$$(extract 32 bytes of entropy from the spending key)
4. $$v \leftarrow \text{randomBytes}(32)$$ (sample another 32 random bytes)
5. $$n \leftarrow \text{SHA512}(x\ ||\ v\ ||\ m) \mod r \in \mathbb{F\_r}$$ ("reduce" hash output into $$\mathbb{F}\_r$$, using extra entropy from $$sk$$ addition to the rng)
6. $$R \leftarrow n \times G$$ (the rest is a "standard" Schnorr signature)
7. $$c \leftarrow H(\text{PK.X}\ ||\ R.X\ ||\ R.Y\ ||\ m)$$
8. $$z \leftarrow n - \text{sk} \cdot c$$
9. The signature is the pair $$(c, z)$$

To verify, we take as input the message $$m$$ and signature $$(c, z)$$ and do the following (this is the standard verification procedure):

1. $$Z \leftarrow z \times G$$
2. $$P \leftarrow c \times \text{PK}$$
3. $$R \leftarrow Z + P$$
4. $$cp \leftarrow H(\text{PK.X}\ ||\ R.X\ ||\ R.Y\ ||\ m)$$
5. Accept the signature if $$cp = c$$. Reject it otherwise.

### Security Requirements

The signature scheme must be strongly unforgeable under chosen-message-attack (SUF-CMA). 

### Security Argument

Schnorr signature schemes are known to be secure under the discrete log assumption and the random-oracle model. In our case, it's secure based on the following assumptions:

1. Poseidon is a random oracle
2. The discrete log problem is infeasible in Baby Jubjub


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